A charged capacitro and an inductor are connected in series. At time t = 0 , the current is zero but the capacitor is charged, If T is the period of resulting oscillations, then the time after which current in the circuit becomes maximum , is

Options

A. T
B. `(T)/(4)`
C. `(T)/(2)`
D. `(T)/(6)`

Answer

To solve the problem, we need to analyze the behavior of a series LC circuit consisting of a charged capacitor and an inductor. The key points to consider are the energy oscillation between the capacitor and the inductor and the relationship between time and current in this oscillating system.

### Step-by-Step Solution:

1. **Understanding Initial Conditions**:
- At time \( t = 0 \), the capacitor is fully charged, and the current in the circuit is zero. This means all the energy is stored in the capacitor as electric potential energy.

2. **Energy in the Capacitor**:
- The energy stored in the capacitor at this moment can be expressed as:
\[ U_C = \frac{Q^2}{2C} \]
where \( Q \) is the charge on the capacitor and \( C \) is the capacitance.

3. **Oscillation in the LC Circuit**:
- The energy in the circuit oscillates between the capacitor and the inductor. As the capacitor discharges, the current starts to flow through the inductor, and energy is transferred to the inductor as magnetic energy.

4. **Current and Maximum Energy Transfer**:
- The current in the circuit reaches its maximum value when all the energy has been transferred from the capacitor to the inductor. This occurs after a quarter of the oscillation period, \( T/4 \).

5. **Period of Oscillation**:
- The period \( T \) of the oscillation in an LC circuit is given by:
\[ T = 2\pi\sqrt{LC} \]
where \( L \) is the inductance.

6. **Time for Maximum Current**:
- Since the current is zero at \( t = 0 \) and reaches its maximum at \( t = T/4 \), we conclude that the time after which the current in the circuit becomes maximum is:
\[ t = \frac{T}{4} \]

### Final Answer:
The time after which the current in the circuit becomes maximum is \( \frac{T}{4} \).

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